On the comparison of stable and unstable $p$-completion

TL;DR

This paper characterizes when a p-complete nilpotent space has a p-complete suspension spectrum based on the boundedness of its homotopy groups, revealing complex behaviors in the presence of unbounded p-torsion.

Contribution

It provides a homological criterion for p-completeness of spectra and explores the stable homotopy groups of Eilenberg–Mac Lane spaces using Goodwillie calculus.

Findings

p-complete nilpotent space has a p-complete suspension spectrum iff its homotopy groups are bounded p-torsion

Unbounded p-torsion in homotopy groups leads to uncountable rational vector spaces in homology and stable homotopy groups

Established a homological criterion for p-completeness of connective spectra

Abstract

In this note we show that a $p$-complete nilpotent space $X$ has a $p$-complete suspension spectrum if and only if its homotopy groups $\pi_*X$ are bounded $p$-torsion. In contrast, if $\pi_*X$ is not all bounded $p$-torsion, we locate uncountable rational vector spaces in the integral homology and in the stable homotopy groups of $X$. To prove this, we establish a homological criterion for $p$-completeness of connective spectra. Moreover, we illustrate our results by studying the stable homotopy groups of $K(\mathbb{Z}_p,n)$ via Goodwillie calculus.